Question 126
Determine all integers $n \geqslant 3$ for which there exist $n$ polynomials $P_{1}, \ldots, P_{n}$ with real coefficients such that, for all distinct $i, j, k$: $P_{i}+P_{j}$ has no real root and $P_{i}+P_{j}+P_{k}$ has at least one real root.